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%% section 9 Moments of Nucleons and Nuclei in the Light-Cone Formalism [slac-pub-7056-0-0-9 in slac-pub-7056-0-0-9: slac-pub-7056-0-0-10]
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\section{\usemenu{slac-pub-7056::context::slac-pub-7056-0-0-9}{Moments of Nucleons and Nuclei in the Light-Cone Formalism}}\label{section::slac-pub-7056-0-0-9}
The use of covariant kinematics leads to a number of striking
conclusions for the electromagnetic and weak moments of nucleons and
nuclei. For example, magnetic moments cannot be written as the naive
sum $\overrightarrow\mu = \sum\overrightarrow\mu_i$ of the magnetic
moments of the constituents, except in the nonrelativistic limit
where the radius of the bound state is much larger than its Compton
scale: $R_A M_A\gg 1$. The deuteron quadrupole moment is in general
nonzero even if the nucleon-nucleon bound state has no $D$-wave
component \cite{94}. Such effects are due to the fact that even
``static'' moments must be computed as transitions between states of
different momentum $p^\mu$ and $p^\mu + q^\mu$, with $q^\mu
\rightarrow 0$. Thus one must construct current matrix elements
between boosted states. The Wigner boost generates nontrivial
corrections to the current interactions of bound systems
\cite{95}. Remarkably, in the case of the deuteron, both the
quadrupole and magnetic moments become equal to that of the Standard
Model in the limit $M_d R_d\rightarrow 0.$ In this limit, the three
form factors of the deuteron have the same ratios as do those of the
$W$ boson in the Standard Model \cite{94}.
One can also use light-cone methods to show that the proton's
magnetic moment $\mu_p$ and its axial-vector coupling $g_A$ have a
relationship independent of the specific form of the light-cone
wavefunction \cite{96}. At the physical value of the proton
radius computed from the slope of the Dirac form factor, $R_1=0.76$
fm, one obtains the experimental values for both $\mu_p$ and $g_A$;
the helicity carried by the valence $u$ and $d$ quarks are each
reduced by a factor $\simeq 0.75$ relative to their nonrelativistic
values. At infinitely small radius $R_p M_p\rightarrow 0$, $\mu_p$
becomes equal to the Dirac moment, as demanded by the
Drell-Hearn-Gerasimov sum rule \cite{97,98}. Another
surprising fact is that as $R_1 \rightarrow 0$ the constituent quark
helicities become completely disoriented and $g_A \rightarrow 0$.
One can understand the origins of the above universal features even
in an effective three-quark light-cone Fock description of the
nucleon. In such a model, one assumes that additional degrees of
freedom (including zero modes) can be parameterized through an
effective potential \cite{38}. After truncation, one could in
principle obtain the mass $M$ and light-cone wavefunction of the
three-quark bound-states by solving the Hamiltonian eigenvalue
problem. It is reasonable to assume that adding more quark and
gluonic excitations will only refine this initial approximation
\cite{10}. In such a theory the constituent quarks will also
acquire effective masses and form factors.
Since we do not have an explicit representation for the effective
potential in the light-cone Hamiltonian $P^-_{\rm eff}$ for three
quarks, we shall proceed by making an Ansatz for the momentum-space
structure of the wavefunction $\Psi$. Even without explicit
solutions of the Hamiltonian eigenvalue problem, one knows that the
helicity and flavor structure of the baryon eigenfunctions will
reflect the assumed global SU(6) symmetry and Lorentz invariance of
the theory. As we will show below, for a given size of the proton
the predictions and interrelations between observables at $Q^2=0,$
such as the proton magnetic moment $\mu_p$ and its axial coupling
$g_A,$ turn out to be essentially independent of the shape of the
wavefunction \cite{96}.
The light-cone model given in Ref. \cite{99} provides a
framework for representing the general structure of the effective
three-quark wavefunctions for baryons. The wavefunction $\Psi$ is
constructed as the product of a momentum wavefunction, which is
spherically symmetric and invariant under permutations, and a
spin-isospin wave function, which is uniquely determined by
SU(6)-symmetry requirements. A Wigner-Melosh rotation
\cite{100,101} is applied to the spinors, so that the
wavefunction of the proton is an eigenfunction of $J$ and $J_z$ in
its rest frame \cite{102,103,104}. To represent the range
of uncertainty in the possible form of the momentum wavefunction,
one can choose two simple functions of the invariant mass ${\cal M}$
of the quarks:
\begin{eqnarray}
\psi_{\rm H.O.}({\cal M}^2) &=& N_{\rm H.O.}\exp(-{\cal
M}^2/2\beta^2),\\ \psi_{\rm Power}({\cal M}^2) &=& N_{\rm Power}
(1+{\cal M}^2/\beta^2)^{-p}\; ,
\end{eqnarray}
where $\beta$ sets the characteristic internal momentum scale.
Perturbative QCD predicts a nominal power-law fall off at large
$k_\perp$ corresponding to $p=3.5$ \cite{38}. The Melosh rotation
insures that the nucleon has $j=\ha$ in its rest system. It has the
matrix representation \cite{101}
\begin{equation}
R_M(x_i,k_{\perp i},m)={m+x_i {\cal M}-i\overrightarrow
\sigma\cdot(\vec n\times\vec k_i)\over\sqrt{(m+x_i {\cal M})^2+
k_{\perp i}^2} }
\end{equation}
with $\vec n=(0,0,1)$, and it becomes the unit matrix if the quarks
are collinear, $R_M(x_i,0,m)=1.$ Thus the internal transverse
momentum dependence of the light-cone wavefunctions also affects its
helicity structure \cite{95}.
The Dirac and Pauli form factors $F_1(Q^2)$ and $F_2(Q^2)$ of the
nucleons are given by the spin-conserving and the spin-flip matrix
elements of the vector current $J^+_V$ (at $Q^2=-q^2$) \cite{15}
\begin{eqnarray}
F_1(Q^2) &=& \langle p+q,\uparrow | J^+_V |
p,\uparrow \rangle , \\
(Q_1-i Q_2) F_2(Q^2) &=& -2M\langle
p+q,\uparrow | J^+_V | p, \downarrow \rangle \; .
\end{eqnarray}
We then can calculate the anomalous magnetic moment
$a=\lim_{Q^2\rightarrow 0} F_2(Q^2)$.\footnote{The total proton
magnetic moment is $\mu_p = {e \over 2M}(1+a_p).$} The same
parameters as in Ref. \cite{99} are chosen, namely $m=0.263$
GeV (0.26 GeV) for the up (down) quark masses, $\beta=0.607$ GeV
(0.55 GeV) for $\psi_{\rm Power}$ ($\psi_{\rm H.O.}$), and $p=3.5$.
The quark currents are taken as elementary currents with Dirac
moments ${e_q \over 2 m_q}.$ All of the baryon moments are well-fit
if one takes the strange quark mass as 0.38 GeV. With the above
values, the proton magnetic moment is 2.81 nuclear magnetons, and
the neutron magnetic moment is $-1.66$ nuclear magnetons. (The
neutron value can be improved by relaxing the assumption of isospin
symmetry.) The radius of the proton is 0.76 fm, i.e., $M_p R_1=3.63$.
In Fig.~3(a) we show the functional relationship between the
anomalous moment $a_p$ and its Dirac radius predicted by the
three-quark light-cone model. The value of
\begin{equation}
R^2_1 = -6 {dF_1(Q^2)\over dQ^2}\Bigl\vert_{Q^2=0}
\end{equation}
is varied by changing $\beta$ in the light-cone wavefunction while
keeping the quark mass $m$ fixed. The prediction for the power-law
wavefunction $\psi_{\rm Power}$ is given by the broken line; the
continuous line represents $\psi_{\rm H.O.}$. Figure~3(a) shows
that when one plots the dimensionless observable $a_p$ against the
dimensionless observable $M R_1$ the prediction is essentially
independent of the assumed power-law or Gaussian form of the
three-quark light-cone wavefunction. Different values of $p>2$ also
do not affect the functional dependence of $a_p(M_p R_1)$ shown in
Fig.~3(a). In this sense the predictions of the three-quark
light-cone model relating the $Q^2 \rightarrow 0$ observables are
essentially model-independent. The only parameter controlling the
relation between the dimensionless observables in the light-cone
three-quark model is $m/M_p$ which is set to 0.28. For the physical
proton radius $M_p R_1=3.63$ one obtains the empirical value for
$a_p=1.79$ (indicated by the dotted lines in Fig. 3(a)).
\setcounter{footnote}{0}
The prediction for the anomalous moment $a$ can be written
analytically as $a=\langle \gamma_V \rangle a^{\rm NR}$, where
$a^{\rm NR}=2M_p/3m$ is the nonrelativistic ($R\rightarrow\infty$)
value and $\gamma_V$ is given as \cite{105}
\begin{equation}
\gamma_V(x_i,k_{\perp i},m)=
{3m\over {\cal M}}\left[ {(1-x_3){\cal M}(m+x_3 {\cal
M})- \vec k_{\perp 3}^2/2\over (m+x_3 {\cal M})^2+\vec k_{\perp
3}^2}\right]\; .
\end{equation}
The expectation value $\langle \gamma_V \rangle$ is evaluated
as\footnote{Here $[d^3k]\equiv d\vec k_1d\vec k_2d\vec
k_3\delta(\vec k_1+\vec k_2+ \vec k_3)$. The third component of
$\vec k$ is defined as $k_{3i}\equiv{1\over2}(x_i{\cal M}-{m^2+\vec
k_{\perp i}^2\over x_i {\cal M}})$. This measure differs from the
usual one used in Ref. \cite{38} by the Jacobian $\prod
{dk_{3i}\over dx_i}$ which can be absorbed into the wavefunction.}
\begin{equation}
\langle\gamma_V\rangle = {\int [d^3k] \gamma_V |\psi|^2\over \int
[d^3k] |\psi|^2}\; .
\end{equation}
Let us now take a closer look at the two limits $R \rightarrow
\infty$ and $R\rightarrow 0$. In the nonrelativistic limit we let
$\beta \rightarrow 0$ and keep the quark mass $m$ and the proton
mass $M_p$ fixed. In this limit the proton radius $R_1 \rightarrow
\infty$ and $a_p \rightarrow 2M_p/3m = 2.38$, since $\langle
\gamma_V \rangle \rightarrow 1$.\footnote{This differs slightly from
the usual nonrelativistic formula $1+a=\sum_q {e_q\over e} {M_p\over
m_q}$ due to the nonvanishing binding energy which results in $M_p
\neq 3m_q$.} Thus the physical value of the anomalous magnetic
moment at the empirical proton radius $M_p R_1=3.63$ is reduced by
25\% from its nonrelativistic value due to relativistic recoil and
nonzero $k_\perp$.\footnote{The nonrelativistic value of the neutron
magnetic moment is reduced by 31\%.}
\begin{figure}
\centerline{\epsfbox{8084A04.eps}}
%%SJB new caption
\caption{(a).
The anomalous magnetic moment of the proton $a_p=F_2(0)$ as a
function of its Dirac radius $M_p R_1 $ in Compton units. (b). The
axial vector coupling of the neutron to proton beta-decay as a
function of $M_p R_1.$ In each figure, the broken line is computed
from a wavefunction with power-law falloff and the solid curve is
computed from a gaussian wavefunction. The experimental values at
the physical proton Dirac radius are indicated by the dotted line.
(From Ref. \ref{bs94}.)}
\label{fig3}
\end{figure}
To obtain the ultra-relativistic limit we let $\beta \rightarrow
\infty$ while keeping $m$ fixed. In this limit the proton becomes
pointlike, $M_p R_1 \rightarrow 0$, and the internal transverse
momenta $k_\perp \rightarrow\infty$. The anomalous magnetic momentum
of the proton goes linearly to zero as $a=0.43 M_p R_1$ since
$\langle\gamma_V\rangle\rightarrow 0$. Indeed, the
Drell-Hearn-Gerasimov sum rule \cite{97,98} demands that
the proton magnetic moment become equal to the Dirac moment at small
radius. For a spin-${1\over2}$ system
\begin{equation}
a^2={M^2\over 2\pi^2\alpha}\int_{s_{th}}^\infty {ds\over s}\left[
\sigma_P(s)-\sigma_A(s)\right]\; ,
\end{equation}
where $\sigma_{P(A)}$ is the total photoabsorption cross section
with parallel (antiparallel) photon and target spins. If we take the
point-like limit, such that the threshold for inelastic excitation
becomes infinite while the mass of the system is kept finite, the
integral over the photoabsorption cross section vanishes and $a=0$
\cite{15}. In contrast, the anomalous magnetic moment of the
proton does not vanish in the nonrelativistic quark model as
$R\rightarrow 0$. The nonrelativistic quark model does not reflect
the fact that the magnetic moment of a baryon is derived from lepton
scattering at nonzero momentum transfer, i.e., the calculation of a
magnetic moment requires knowledge of the boosted wavefunction. The
Melosh transformation is also essential for deriving the DHG sum
rule and low-energy theorems of composite systems \cite{95}.
A similar analysis can be performed for the axial-vector coupling
measured in neutron decay. The coupling $g_A$ is given by the
spin-conserving axial current $J_A^+$ matrix element
\begin{equation}
g_A(0) =\langle p,\uparrow | J^+_A | p,\uparrow \rangle\; .
\end{equation}
The value for $g_A$ can be written as $g_A=\langle \gamma_A \rangle
g_A^{\rm NR}$, with $g_A^{\rm NR}$ being the nonrelativistic value of
$g_A$ and with $\gamma_A$ given by \cite{105,106}
\begin{equation}
\gamma_A(x_i,k_{\perp i},m)={(m+x_3 {\cal M})^2-
k_{\perp 3}^2\over (m+x_3 {\cal
M})^2+ k_{\perp 3}^2}\; .
\label{gammaa}
\end{equation}
In Fig.~3(b) the axial-vector coupling is plotted against the proton
radius $M_p R_1$. The same parameters and the same line
representation as in Fig.~3(a) are used. The functional dependence
of $g_A(M_p R_1)$ is also found to be independent of the assumed
wavefunction. At the physical proton radius $M_p R_1=3.63$, one
predicts the value $g_A = 1.25$ (indicated by the dotted lines in
Fig.~3(b)), since $\langle \gamma_A \rangle =0.75$. The measured
value is $g_A= 1.2573\pm 0.0028$ \cite{107}. This is a 25\%
reduction compared to the nonrelativistic SU(6) value $g_A=5/3,$
which is only valid for a proton with large radius $R_1 \gg 1/M_p.$
As shown in Ref. \cite{106}, the Melosh rotation generated by the
internal transverse momentum spoils the usual identification of the
$\gamma^+ \gamma_5$ quark current matrix element with the total
rest-frame spin projection $s_z$, thus resulting in a reduction of
$g_A$.
Thus, given the empirical values for the proton's anomalous moment
$a_p$ and radius $M_p R_1,$ its axial-vector coupling is
automatically fixed at the value $g_A=1.25.$ This is an essentially
model-independent prediction of the three-quark structure of the
proton in QCD. The Melosh rotation of the light-cone wavefunction
is crucial for reducing the value of the axial coupling from its
nonrelativistic value 5/3 to its empirical value. The near equality
of the ratios $g_A/g_A(R_1 \rightarrow \infty)$ and $a_p/a_p(R_1
\rightarrow \infty)$ as a function of the proton radius $R_1$ shows
the wave-function independence of these quantities. We emphasize
that at small proton radius the light-cone model predicts not only a
vanishing anomalous moment but also $ \lim_{R_1 \rightarrow 0}
g_A(M_p R_1)=0$. One can understand this physically: in the zero
radius limit the internal transverse momenta become infinite and the
quark helicities become completely disoriented. This is in
contradiction with chiral models, which suggest that for a zero
radius composite baryon one should obtain the chiral symmetry result
$g_A=1$.
The helicity measures $\Delta u$ and $\Delta d$ of the nucleon each
experience the same reduction as does $g_A$ due to the Melosh
effect. Indeed, the quantity $\Delta q$ is defined by the axial
current matrix element
\begin{equation}
\Delta q=\langle p,\uparrow | \bar
q\gamma^+\gamma_5 q | p,\uparrow \rangle\; ,
\end{equation}
and the value for $\Delta q$ can be written analytically as $\Delta
q=\langle \gamma_A \rangle \Delta q^{\rm NR}$, with $\Delta q^{\rm
NR}$ being the nonrelativistic or naive value of $\Delta q$ and
$\gamma_A$ given by Eq. (\docLink{slac-pub-7056-0-0-9.tcx}[gammaa]{28}).
The light-cone model also predicts that the quark helicity sum
$\Delta\Sigma=\Delta u+\Delta d$ vanishes as a function of the
proton radius $R_1$. Since $\Delta\Sigma$ depends on the proton
size, it cannot be identified as the vector sum of the rest-frame
constituent spins. As emphasized in Ref. \cite{106}, the rest-frame
spin sum is not a Lorentz invariant for a composite system.
Empirically, one can measure $\Delta q$ from the first moment of the
leading-twist polarized structure function $g_1(x,Q).$ In the
light-cone and parton model descriptions, $\Delta q=\int_0^1 dx
[q^\uparrow (x) - q^\downarrow (x)]$, where $q^\uparrow (x)$ and
$q^\downarrow (x)$ can be interpreted as the probability for finding
a quark or antiquark with longitudinal momentum fraction $x$ and
polarization parallel or antiparallel to the proton helicity in the
proton's infinite momentum frame \cite{38}. [In the infinite
momentum frame there is no distinction between the quark helicity
and its spin projection $s_z.$] Thus $\Delta q$ refers to the
difference of helicities at fixed light-cone time or at infinite
momentum; it cannot be identified with
$q(s_z=+{1\over2})-q(s_z=-{1\over2}),$ the spin carried by each
quark flavor in the proton rest frame in the equal-time formalism.
Thus the usual SU(6) values $\Delta u^{\rm NR}=4/3$ and $\Delta
d^{\rm NR}=-1/3$ are only valid predictions for the proton at large
$M R_1.$ At the physical radius the quark helicities are reduced by
the same ratio 0.75 as is $g_A/g_A^{\rm NR}$ due to the Melosh
rotation. Qualitative arguments for such a reduction have been given
in Refs. \cite{109,110}. For $M_p R_1 = 3.63,$ the
three-quark model predicts $\Delta u=1,$ $\Delta d=-1/4,$ and
$\Delta\Sigma=\Delta u+\Delta d = 0.75$. Although the gluon
contribution $\Delta G=0$ in our model, the general sum rule
\cite{111}
\begin{equation}
{1\over2}\Delta\Sigma +\Delta G+L_z= {1\over2}
\end{equation}
is still satisfied, since the Melosh transformation effectively
contributes to $L_z$.
Suppose one adds polarized gluons to the three-quark light-cone
model. Then the flavor-singlet quark-loop radiative corrections to
the gluon propagator will give an anomalous contribution $\delta
(\Delta q)=-{\alpha_s\over2\pi}\Delta G$ to each light quark
helicity \cite{112}. The predicted value of $g_A = \Delta
u-\Delta d$ is of course unchanged. For illustration we shall choose
${\alpha_s\over2\pi}\Delta G=0.15$. The gluon-enhanced quark model
then gives the values in Table~1, which agree well with the present
experimental values. Note that the gluon anomaly contribution to
$\Delta s$ has probably been overestimated here due to the large
strange quark mass. One could also envision other sources for this
shift of $\Delta q$ such as intrinsic flavor \cite{110}. A
specific model for the gluon helicity distribution in the nucleon
bound state is given in Ref. \cite{17}.
\begin{table}
\begin{center}
\begin{tabular}{|c|c|c|c|c|}
\hline Quantity & NR & $3q$ & $3q+g$ & Experiment \\ \hline $\Delta u$
& ${4\over3}$ & 1 & 0.85 & $0.83\pm 0.03 $ \\ $\Delta d$ &$-{1\over3}$
& $-{1\over4}$ & --0.40 & $-0.43\pm 0.03 $\\ $\Delta s$ & 0 & 0 &
--0.15 & $-0.10\pm 0.03 $\\ $\Delta \Sigma$ &1 & ${3\over4}$ & 0.30 &
$0.31\pm 0.07 $\\ \hline
\end{tabular}
\end{center}
\caption{Comparison of the quark content of the proton
in the nonrelativistic quark model (NR), in the three-quark model
($3q$), in a gluon-enhanced three-quark model ($3q+g$), and with
experiment [112]. %\cite{ek94}.
}
\end{table}
In the above analysis of the singlet moments, it is assumed that all
contributions to the sea quark moments derive from the gluon anomaly
contribution $\delta (\Delta q)=-{\alpha_s\over2\pi}\Delta G$. In
this case the strange and anti-strange quark distributions will be
identical. On the other hand, if the strange quarks derive from the
intrinsic structure of the proton, then one would not expect this
symmetry. For example, in the intrinsic strangeness wavefunctions,
the dominant fluctuations in the nucleon wavefunction are most
likely dual to intermediate $\Lambda$-$K$ configurations since they
have the lowest off-shell light-cone energy and invariant mass. In
this case $s(x)$ and $\bar s(x)$ will be different.
The light-cone formalism also has interesting consequences for spin
correlations in jet fragmentation. In LEP or SLC one produces $s$
and $\bar s$ quarks with opposite helicity. This produces a
correlation of the spins of the $\Lambda$ and $\overline\Lambda$,
each produced with large $z$ in the fragmentation of their
respective jet. The $\Lambda$ spin essentially follows the spin of
the strange quark since the $ud$ has $J=0$. However, this cannot be
a 100\% correlation since the $\Lambda$ generally is produced with
some transverse momentum relative to the $s$ jet. In fact, from the
light-cone analysis of the proton spin, we would expect no more than
a 75\% correlation since the $\Lambda$ and proton radius should be
almost the same. On the other hand if $z=E_\Lambda/E_s \to 1,$ there
can be no wasted energy in transverse momentum. At this point one
could have 100\% polarization. In fact, the nonvalence Fock states
will be suppressed at the extreme kinematics, so there is even more
reason to expect complete helicity correlation in the endpoint
region.
We can also apply a similar idea to the study of the fragmentation
of strange quarks to $\Lambda$s produced in deep inelastic lepton
scattering on a proton at ELFE. One could use the correlation
between the spin of the target proton and the spin of the $\Lambda$
to directly measure the strange polarization $\Delta s.$ It is
conceivable that any differences between $\Delta s$ and $\Delta \bar
s$ in the nucleon wavefunction could be distinguished by measuring
the correlations between the target polarization and the $\Lambda$
and $\overline\Lambda$ polarization in deep inelastic lepton proton
collisions at ELFE or in the target polarization region in
hadron-proton collisions.
In summary, we have shown that relativistic effects are crucial for
understanding the spin structure of nucleons. By plotting
dimensionless observables against dimensionless observables, we
obtain relations that are independent of the momentum-space form of
the three-quark light-cone wavefunctions. For example, the value of
$g_A \simeq 1.25$ is correctly predicted from the empirical value of
the proton's anomalous moment. For the physical proton radius $M_p
R_1= 3.63$, the inclusion of the Wigner-Melosh rotation due to the
finite relative transverse momenta of the three quarks results in a
$\sim 25\%$ reduction of the nonrelativistic predictions for the
anomalous magnetic moment, the axial vector coupling, and the quark
helicity content of the proton. At zero radius, the quark
helicities become completely disoriented because of the large
internal momenta, resulting in the vanishing of $g_A$ and the total
quark helicity $\Delta \Sigma.$
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